ΣΕΛΙΔΑ:    ΆΣΚΗΣΗ   ΜΗΝΑ.   --   Μήνας:   Μάρτιος 2016

\boldsymbol{\color{Black} \acute{A}\sigma\kappa\eta\sigma\eta \:\:4/2016.}

\boldsymbol{\color{Blue} \bullet \,\,\, N\alpha \,\, \delta\epsilon\acute{\iota}\xi\epsilon\tau\epsilon\,\, \acute{o}\tau\iota :\,\,\, K\acute{\alpha}\theta\epsilon\,\, \sigma\upsilon\nu\acute{\alpha}\rho\tau\eta\sigma\eta \quad f\,:\,IR\,\,\rightarrow \,\,IR \quad \gamma\rho\acute{\alpha}\phi\epsilon\tau\alpha\iota \,\, \omega\varsigma}

\boldsymbol{\color{Blue} \acute{\alpha}\theta\rho o \iota\sigma\mu\alpha \,\, \mu\iota\alpha\varsigma\,\,\, \acute{\alpha}\rho\tau\iota\alpha\varsigma \,\, \kappa\alpha\iota \,\, \mu\iota\alpha\varsigma \,\, \pi\epsilon\rho\iota\tau\tau\acute{\eta}\varsigma \,\, \sigma\upsilon\nu\acute{\alpha}\rho\tau\eta\sigma\eta\varsigma.}

\boldsymbol{\color{Blue} \bullet \,\,\,N\alpha \,\, \beta\rho\epsilon\acute{\iota}\tau\epsilon \,\, \sigma\upsilon\nu\acute{\alpha}\rho\tau\eta\sigma\eta \quad f\,: \: IR\,\,\rightarrow \,\,IR \quad \mu\epsilon : \: f(x)f(-x)=4 \quad \gamma\iota\alpha \: \kappa\acute{\alpha}\theta\epsilon \: x \in IR.}

\boldsymbol{\color{black} A\pi\acute{o}\delta\epsilon\iota\xi\eta }

\boldsymbol{\color{Blue} \bullet \:\:\: \Upsilon\pi o \theta\acute{\epsilon}\tau o \upsilon\mu\epsilon \,\,\, \acute{o}\tau\iota \,\,\, \upsilon\pi\acute{\alpha}\rho\chi o\upsilon\nu \,\,\, \sigma\upsilon\nu\alpha\rho\tau\acute{\eta}\sigma\epsilon\iota\varsigma \,\,\,a,\,\,\,p \,\,:\,IR\,\,\rightarrow \,\,IR \,\,\, \epsilon\kappa \,\,\, \tau\omega\nu \,\,\, o\pi o \acute{\iota}\omega\nu}

\boldsymbol{\color{Blue} \eta \,\,\, a \,\,\, \acute{\alpha}\rho\tau\iota\alpha \,\,\, \kappa\alpha\iota \,\,\, \eta \,\,\, p \,\,\, \pi\epsilon\rho\iota\tau\tau\acute{\eta} \,\,\, \acute{\epsilon}\tau\sigma\iota \,\,\, \acute{\omega}\sigma\tau\epsilon :}

\boldsymbol{\color{Blue} f(x)=a(x)+p(x) \,\,\, \gamma\iota\alpha \: \kappa\acute{\alpha}\theta\epsilon \: x \in IR \,\,\,\,(1).}

\boldsymbol{\color{Blue} E\acute{\iota}\nu\alpha\iota \,\,\, \tau\acute{o} \tau\epsilon: \,\,\, f(-x)=a(-x)+p(-x)=a(x)-p(x) \,\,\,\, (2) \,\,\, \gamma\iota\alpha \: \kappa\acute{\alpha}\theta\epsilon \: x \in IR. }

\boldsymbol{\color{Blue} \Lambda\acute{\upsilon}\nu o \upsilon\mu\epsilon \,\,\, \tau o \,\,\, \sigma\acute{\upsilon}\sigma\tau\eta\mu\alpha \,\,\, \tau\omega\nu \,\,\, (1) \,\,\, \kappa\alpha\iota \,\,\, (2) \,\,\, \omega\varsigma \,\,\, \pi\rho o \varsigma \,\,\, a(x) \,\,\, \kappa\alpha\iota \,\,\, p(x) \,\,\, \kappa\alpha\iota \,\,\, \acute{\epsilon}\chi o \upsilon\mu\epsilon: }

\boldsymbol{\color{Blue} a(x)=\frac{f(x)+f(-x)}{2}, \,\,\,\, p(x)=\frac{f(x)-f(-x)}{2} \,\,\,\, (3).}

\boldsymbol{\color{Blue} \Lambda\acute{o}\gamma\omega \,\,\, \tau\omega\nu \,\,\, \sigma\chi\acute{\epsilon}\sigma\epsilon\omega\nu \,\,\,\, (3) \,\,\,\, \epsilon\acute{\iota}\nu\alpha\iota:}

\boldsymbol{\color{Blue} a(-x)=\frac{f(-x)+f(x)}{2}=a(x) \,\,\,\, \kappa\alpha\iota \,\,\,\, p(-x)=\frac{f(-x)-f(x)}{2}=-p(x).}

\boldsymbol{\color{Blue} E\pi o\mu\acute{\epsilon}\nu\omega\varsigma \,\,\, o\iota \,\,\, \sigma\upsilon\nu\alpha\rho\tau\acute{\eta}\sigma\epsilon\iota\varsigma \,\,\,\, a \,\,\, \kappa\alpha\iota \,\,\,\, p \,\,\, \epsilon\acute{\iota}\nu\alpha\iota \,\,\, \eta \,\,\, \mu\epsilon\nu \,\,\, a \,\,\, \acute{\alpha}\rho\tau\iota\alpha \,\,\, \eta \,\,\, \delta\epsilon \,\,\, p \,\,\, \pi\epsilon\rho\iota\tau\tau\acute{\eta}. }

\boldsymbol{\color{Blue} \acute{A}\rho\alpha \,\,\, \delta o \theta\epsilon\acute{\iota}\sigma\eta\varsigma \,\,\, \sigma\upsilon\nu\acute{\alpha}\rho\tau\eta\sigma\eta\varsigma \,\,\, f\,:\,IR\,\,\rightarrow \,\,IR \,\,\, \upsilon\pi\acute{\alpha}\rho\chi o \upsilon\nu \,\,\, \pi\acute{\alpha}\nu\tau o \tau\epsilon \,\,\, \sigma\upsilon\nu\alpha\rho\tau\acute{\eta}\sigma\epsilon\iota\varsigma}

\boldsymbol{\color{Blue} \,\,\,a,\,\,\,p \,\,:\,IR\,\,\rightarrow \,\,IR \,\,\, \epsilon\kappa \,\,\, \tau\omega\nu \,\,\, o\pi o \acute{\iota}\omega\nu \,\,\, \eta \,\,\, a \,\,\, \acute{\alpha}\rho\tau\iota\alpha \,\,\, \kappa\alpha\iota \,\,\, \eta \,\,\, p \,\,\, \pi\epsilon\rho\iota\tau\tau\acute{\eta} \,\,\, \acute{\epsilon}\tau\sigma\iota \,\,\, \acute{\omega}\sigma\tau\epsilon :}

\boldsymbol{\color{Blue} f(x)=a(x)+p(x) \,\,\, \gamma\iota\alpha \: \kappa\acute{\alpha}\theta\epsilon \: x \in IR.}

\boldsymbol{\color{Blue} E\acute{\iota}\nu\alpha\iota \,\,\, \delta\epsilon: \,\,\,\, a(x)=\frac{f(x)+f(-x)}{2} \,\,\, \kappa\alpha\iota \,\,\, p(x)=\frac{f(x)-f(-x)}{2} \,\,\,\, \gamma\iota\alpha \: \kappa\acute{\alpha}\theta\epsilon \: x \in IR.}

\boldsymbol{\color{Blue} \bullet \:\:\: \acute{E}\sigma\tau\omega \,\,\, \acute{o}\tau\iota \,\,\, \upsilon\pi\acute{\alpha}\rho\chi\epsilon\iota \,\,\, \sigma\upsilon\nu\acute{\alpha}\rho\tau\eta\sigma\eta }

\boldsymbol{\color{Blue} \,\,\, f\,: \: IR\,\,\rightarrow \,\,IR \quad \mu\epsilon : \: f(x)f(-x)=4 \quad \gamma\iota\alpha \: \kappa\acute{\alpha}\theta\epsilon \: x \in IR. }

\boldsymbol{\color{Blue} H \,\,\, f \,\,\, \gamma\rho\acute{\alpha}\phi\epsilon\tau\alpha\iota \,\,\, \omega\varsigma \,\,\, \acute{\alpha}\theta\rho o \iota\sigma\mu\alpha \,\,\, \mu\iota\alpha\varsigma \,\,\, \acute{\alpha}\rho\tau\iota\alpha\varsigma \,\,\, \sigma\upsilon\nu\acute{\alpha}\rho\tau\eta\sigma\eta\varsigma \,\,\, a \,\,:\,IR\,\,\rightarrow \,\,IR }

\boldsymbol{\color{Blue} \,\,\, \kappa\alpha\iota \,\,\, \mu\iota\alpha\varsigma \,\,\, \pi\epsilon\rho\iota\tau\tau\acute{\eta}\varsigma \,\,\, \sigma\upsilon\nu\acute{\alpha}\rho\tau\eta\sigma\eta\varsigma \,\,\, p \,\,:\,IR\,\,\rightarrow \,\,IR.}

\boldsymbol{\color{Blue} \Delta\eta\lambda\alpha\delta\acute{\eta}: \,\,\, f(x)=a(x)+p(x) \,\,\, \gamma\iota\alpha \: \kappa\acute{\alpha}\theta\epsilon \: x \in IR.}

\boldsymbol{\color{Blue} E\acute{\iota}\nu\alpha\iota \,\,\, \tau\acute{o} \tau\epsilon: \,\,\, f(-x)=a(x)-p(x) \,\,\,\, \gamma\iota\alpha \: \kappa\acute{\alpha}\theta\epsilon \: x \in IR \,\,\, \kappa\alpha\iota: }

\boldsymbol{\color{Blue} f(x)f(-x)=\{a(x)+p(x)\}\{a(x)-p(x)\}=a^2(x)-p^2(x) \,\,\,\stackrel{f(x)f(-x)=4}{\Longrightarrow}}

\boldsymbol{\color{Blue} \Rightarrow a^2(x)=4+p^2(x) \, \Rightarrow \, a(x)=\pm \sqrt{4+p^2(x)} \,\,\,\, \Rightarrow \, f(x)=p(x) \pm \sqrt{4+p^2(x)} \,\,\,\,(4).}

\boldsymbol{\color{Blue} \Sigma\upsilon\nu\epsilon\pi\acute{\omega}\varsigma \,\,\, \upsilon\pi\acute{\alpha}\rho\chi o \upsilon\nu \,\,\, \acute{\alpha}\pi\epsilon\iota\rho\epsilon\varsigma \,\,\, \sigma\upsilon\nu\alpha\rho\tau\acute{\eta}\sigma\epsilon\iota\varsigma \,\,\,f\,: \: IR\,\,\rightarrow \,\,IR \,\,\,\mu\epsilon : }

\boldsymbol{\color{Blue} \: f(x)f(-x)=4 \quad \gamma\iota\alpha \: \kappa\acute{\alpha}\theta\epsilon \: x \in IR.}

\boldsymbol{\color{Blue} O\iota \,\,\, \sigma\upsilon\nu\alpha\rho\tau\acute{\eta}\sigma\epsilon\iota\varsigma \,\,\, f \,\,\, \delta\acute{\iota}\delta o \nu\tau\alpha\iota \,\,\, \alpha\pi\acute{o} \,\,\, \tau\eta\nu \,\,\, (4) \,\,\, \gamma\iota\alpha \,\,\, \alpha\upsilon\theta\alpha\acute{\iota}\rho\epsilon\tau\alpha \,\,\, \epsilon\pi\iota\lambda\epsilon\gamma\mu\acute{\epsilon}\nu\eta \,\,\, }

\boldsymbol{\color{Blue} \pi\epsilon\rho\iota\tau\tau\acute{\eta} \,\,\, \sigma\upsilon\nu\acute{\alpha}\rho\tau\eta\sigma\eta \,\,\, p \,\,:\,IR\,\,\rightarrow \,\,IR.}

\boldsymbol{\color{Blue} \Gamma\iota\alpha \,\,\, \pi\alpha\rho\acute{\alpha}\delta\epsilon\iota\gamma\mu\alpha, \,\,\, \mu\epsilon \,\,\, a(x)=x, \,\,\, \epsilon\acute{\iota}\nu\alpha\iota:}

\boldsymbol{\color{Blue} f(x)=x\pm\sqrt{4+x^2} \,\,\, \gamma\iota\alpha \,\,\, \kappa\acute{\alpha}\theta\epsilon \,\,\, x \in IR.}